Maurer–Cartan moduli for non-monotone toric Lagrangians

Determine the space of projective Maurer–Cartan solutions, up to equivalence, for Lagrangian branes in toric varieties so that the disk potential can be defined in the non-monotone setting.

Background

The paper focuses on tropical formulas for disk potentials primarily in the monotone case, where the potential can be defined directly on the space of local systems. In the non-monotone case, the potential is defined on the space of projective Maurer–Cartan solutions (bounding cochains) up to equivalence, reflecting the curved nature of the A-infinity algebra.

However, the relevant moduli of projective Maurer–Cartan solutions is not presently characterized even in toric settings. This gap limits the formulation of clean theorems beyond the monotone case and motivates determining this moduli space in concrete symplectic situations, such as Lagrangian torus fibers in toric varieties.

References

Although the techniques here are not restricted to the monotone case, it is difficult to formulate clean theorems in the non-monotone cases as the disk potential in the monotone case is only defined as a function on the space of projective Maurer-Cartan solutions up to equivalence. But the space of such solutions is unknown, even for toric varieties.

— Tropical disk potential for almost toric manifolds  (2604.03161 - Venugopalan et al., 3 Apr 2026) in Introduction